Fractal dimension Mandelbrot

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

Mandelbrot defines the surface area fractal dimension 8 as... 

Hence plots of log against log Twill have a slope of -D. The parameter D is characteristic of the texture of the particle and was called by Mandelbrot the fractal dimension. The fractal dimension for the outline of a particle lies between 1 and 2, the more irregular the outline, the higher the value. 

Fractals are geometric structures of fractional dimension their theoretical concepts and physical applications were early studied by Mandelbrot [Mandelbrot, 1982]. By definition, any structure possessing a self-similarity or a repeating motif invariant under a transformation of scale is caWcd fractal and may be represented by a fractal dimension. Mathematically, the fractal dimension Df of a set is defined through the relation ... 

The concept of fractal geometry was first introduced by Mandelbrot and it refers to a rough or fragmented geometric shape that is composed of many smaller copies that have the same shape but different sizes of the whole figure and fractal dimension is a statistical tool to measure how the fractal object rills the space (44). 

Fractal Logic This was introduced into fine particles science by Kaye and coworkers (Kaye, op. cit., 1981), who show that the noneuchd-ean logic of Mandelbrot can be applied to describe the ruggedness of a particle profile. A combination of fractal dimension and geometric shape factors such as the aspect ratio can be used to describe a population of fine particles of various shapes, and these can be related to tne functional properties of the particle. 

Fractals are geometric objects, which have a fractal dimension and where the constituent small parts are similar to the whole object. Fractals became well known following the publications of Mandelbrot in 1977 [14,15],... 

Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. 

A concept of mutual transparency or opacity based on the relative evolution of fractal dimension and radius of the clusters has been developed by Mandelbrot [38]. The tendency of fractal systems to interpenetrate is inversely related to the mean number of intersections 2 of two mass fractal objects of size and mass fractal dimensions Dj and D2 placed in the same region of space of dimension d ... 

Many systems exhibit fractal geometries, characterized by structures that look the same on all length scales (Mandelbrot, 1982). Fractal structures can be characterized by a scale law where the number of discrete units is proportional to the dp power of the size of those units, where dp represents the fractal dimension of the structure. 

Image analysis of soil thin sections is the other method that is commonly used for characterizing pore shape. Because of measurement constraints, these analyses are generally conducted in two dimensions, and thus it is the boundary fractal dimension that is used to quantify pore surface roughness (Kampichler Hauser, 1693 Anderson et al., 1996 Pachepsky et al., 1996). The db is defined by the following equation (Mandelbrot, 1983),... 

One of the most exciting developments of the past decade in the study of colloidal silica is the application of the fractal approach to the study of sols and gels. Fractals are disordered systems for which disorder can be described in terms of nonintegral dimension. The concept of fractal geometry, developed by Mandelbrot (II) in the early 1980s, provides a means of quantitatively describing the average structure of certain random objects. The fractal dimension of an object of mass M and radius r is defined by the relation... 

The surface area, pore volume, and pore size of the deposited film depend on such factors as the size and structure (fractal dimension) of the entrained inorganic species, the relative rates of condensation and evaporation during deposition, and the magnitude of the capillary pressure (122). The fractal dimension influences porosity through steric control. Mandelbrot (47) showed that if two objects of radius R are placed independently in the same region of space, the number of intersections (Mi,2) is expressed as... 

Mandelbrot (24) has suggested a different approach to surface irregularity by using fractal dimensions. A recent proliferation of studies has substantiated the hypothesis of self similarity for a number of natural systems ( 71-73V By this approach, surface irregularity scaling is given by the fractal dimension D, whose range is defined as 2 < D < 3 and which is related to the surface area by the proportionality... 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

Since the 1980s, the fractal theory (Mandelbrot 1983) has been fairly applied in avariety of research areas including geology and geophysics. Some Chinese scholars explored the relationship between the fractal characteristics of coal mine area faults and gas outburst (He et al. 2002). They quantified the fault complexity using the fractal dimension as the... 

The value a = 1 corresponds to ideal capacitive behavior. The so-called fractal dimension D introduced by Mandelbrot [37] is a quantity that attains a value between... 

At the same time that he coined the term fractal , Mandelbrot [4] pointed out that fractal dimensions would not suffice to provide a satisfactory description of the geometry of lacunar fractals, and that at least one other parameter, which he termed lacunarity , would be necessary. The key reason for this requirement is vividly illustrated by the fact that Sierpinski carpets (Figure 2.15) with greatly different appearances can have precisely the same fractal (similarity) dimension. Therefore, the fractal dimension alone is not a very reliable diagnostic of the geometry and properties of lacunar fractals. For physical objects, such as porous media, where the geometry of interstices and pores influences a wide range of properties, this means that any attempt to find a unique relationship between the fractal dimension of these objects and, for example, their transport or dielectric properties is most probably doomed to failure, unless one also takes lacunarity explicitly into account. 

As with the concept of fractals itself, a certain level of vagueness characterizes the definition of the fractal dimension. The approach advocated by Mandelbrot in 1975 [1], and reiterated in his 1982 book, is to use the expression fractal dimension as a generic term applicable to all the variants described in Section 2.3, and to use in each specific case whichever definition is most appropriate. This suggestion is adopted by a number of authors [e.g. 56]. However, it could, potentially, lead to considerable confusion if it is followed inconsistently, particularly in cases where different dimensions assume different values (see Section 2.3 for examples). Therefore, many mathematicians consider it safer to refer to specific dimensions by name, such as the correlation dimension, instead of using the generic term fractal dimension [e.g. 5]. 

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